Prior techniques and arrangements for converting asynchronous digital transmission signals to synchronous digital transmission signals, and vice versa, typically required a so-called "clock smoothing phase locked loop" and an associated elastic store unit to smooth gaps in a demultiplexed digital signal, for example, the DS1 digital signal. The smooth signal is then converted to a desired digital signal format, for example, the SONET VT signal, by employing an additional synchronizing elastic store unit and associated phase detector which generates a stuff control signal for the desired format. One such prior arrangement is shown in FIG. 1 and described below. Thus, when converting between, for example, the DS3 digital transmission signal and a synchronous digital transmission signal, for example, the SONET STS1 digital transmission signal using the prior techniques and arrangements would require a large number, for example, 56 phase locked loops and 56 associated elastic store units, to realize the clock smoothing. Additionally, the prior techniques and arrangements which required the use of a clock smoothing additional elastic store unit introduced an unnecessary time delay through the smoothing process.